Baghdad -- The top Pentagon official arrived in Baghdad on Tuesday to consult with American military commanders and Iraqi government leaders on the future U.S. troop presence in Iraq, just a week after President Trump angered Iraq's leaders by suggesting a change in the U.S. mission there. Pat Shanahan, the acting secretary of defense, said before his unannounced trip that he wanted to hear first-hand about the state of Iraq's fight against remnants of the Islamic State of Iraq and Syria (ISIS). Shanahan, who is on his first visit to Iraq, was also to meet Prime Minister Adel Abdul-Mahdi.

In remarks to reporters after leaving Washington on Sunday, Shanahan declined to say whether he would propose that additional U.S. special operations troops be brought to Iraq to, in effect, compensate for a pullout from Syria to begin within weeks.

The U.S. still has about 5,200 troops in Iraq to train and advise its security forces, 16 years after the U.S. invaded to topple Saddam Hussein.

The U.S. mission in Iraq

President Donald Trump upset Iraqis by saying earlier this month that U.S. forces should use their Iraqi positions to keep an eye on neighboring Iran. That is not the stated U.S. mission in Iraq, and Iraqi officials have said Mr. Trump's proposal would violate the Iraqi constitution.

Mr. Trump also has angered Iraqi politicians by arguing that he would keep U.S. troops in Iraq and use the country as a base from which to strike extremists in Syria if necessary, after the 2,000 troops now in Syria depart in coming weeks.

Curbing foreign influence has become a hot-button issue in Iraq after parliamentary elections last year in which Shiite politicians backed by Iran made significant gains. Meanwhile, Shiite militias that fought alongside U.S.-backed Iraqi government troops against ISIS in recent years, gained outsized influence along the way.

This political tension formed the backdrop to Shanahan's visit, which marks his first time in Iraq. He took over as the acting Pentagon chief after Jim Mattis resigned as defense secretary in December. It's unclear whether Mr. Trump will nominate Shanahan for Senate confirmation.

Trump on fight against ISIS, risk of "endless wars"

Mr. Trump has not publicly called for a U.S. withdrawal from Iraq, although he often calls the 2003 invasion a colossal mistake and has said the U.S. should have taken Iraq's oil as compensation for getting rid of Saddam.

Next door in Syria, the fight isn't over

Since the height of its self-proclaimed caliphate that included a third of both Iraq and Syria, ISIS-held territory has now shrunk to a sliver of territory in eastern Syria where the remaining militants are fighting back, hard.

CBS News correspondent Charlie D'Agata has been on the front lines of that fight, alongside the largely Kurdish fighters of the U.S.-backed Syrian Democratic Forces (SDF).

Remaining ISIS fighters hold steady in Syria

As D'Agata reported on Monday, the American-backed forces are calling it the "final push," but the ISIS holdouts were pushing back, and it could prove to be a drawn out, and bloody battle. The militants launched a fierce counterattack on Monday, actually retaking territory from the SDF that D'Agata had visited only days earlier.

An SDF spokesman told CBS News that ISIS may only have 500-600 fighters left in the enclave, but they are among the most battle-hardened and determined of the once huge ISIS contingent.

On Monday the chief military spokesman in the region told D'Agata there are thought to be far more civilians trapped inside the village still held by ISIS than previously thought. They initially estimated that about 1,500 people were trapped there, but D'Agata was told about that many people came flooding out on Monday alone.

One of the SDF commanders leading the fight told D'Agata that he was also worried about what would happen if or when U.S. forces do leave Syria. He warned that the militants had already established an underground network of fighters across the country, and that they were already preparing to plan attacks, and to regroup.

On Monday, a UN report found the terror group remains a threat. In a statement, Under Secretary-General Vladimir Voronkov said despite a fall in "international attacks" in 2018, "this threat is increased by returning, relocating or released foreign terrorist fighters." Voronkov also said the central leadership of ISIS maintains an "intent to generate internationally-directed attacks."

Even if ISIS territory is reclaimed, is U.S. leaving job "halfway" done?